Question

Find the value of $4\cos {60^0}\sin {30^0}$

Answer 1

Hint: Here we are asked to find the value of the given expression that has trigonometric functions. As the expression contains trigonometric functions, we first need to simplify them if needed or we can directly apply the values of those trigonometric ratios by using the trigonometric ratio table then simplify them to find the required value of that expression.

Complete step by step answer:
Since from given that $4\cos {60^0}\sin {30^0}$
To solve the given we must need to know about the sine and cosine tables in the trigonometric identities.
Starting with the trigonometric table of sine and cosine to solve the given problem.
Let us see the sine table in the trigonometric functions with respect to the corresponding angles

Angle in degrees	\[0^\circ \]	\[30^\circ \]	\[45^\circ \]	\[60^\circ \]	\[90^\circ \]
\[\sin \]	\[0\]	\[\dfrac{1}{2}\]	\[\dfrac{1}{{\sqrt 2 }}\]	\[\dfrac{{\sqrt 3 }}{2}\]	\[1\]

Similarly, cosine table in the trigonometric functions with respect to the corresponding angles

Angle in degrees	\[0^\circ \]	\[30^\circ \]	\[45^\circ \]	\[60^\circ \]	\[90^\circ \]
\[\cos \]	\[1\]	\[\dfrac{{\sqrt 3 }}{2}\]	\[\dfrac{1}{{\sqrt 2 }}\]	\[\dfrac{1}{2}\]	\[0\]

Now take the first given equality value, which is $\cos {60^0}$ and we can clearly see that $\cos {60^0} = \dfrac{1}{2}$
Now take the second value from the given, that is $\sin {30^0}$ and we can clearly see that $\sin {30^0} = \dfrac{1}{2}$
Since both the value from the trigonometric identities having the same number as $\dfrac{1}{2}$
Now it was easy to solve the given $4\cos {60^0}\sin {30^0}$
since we know that $\cos {60^0} = \dfrac{1}{2}$ and $\sin {30^0} = \dfrac{1}{2}$
hence applying the values in the given problem and then we get $4\cos {60^0}\sin {30^0} = 4 \times \dfrac{1}{2} \times \dfrac{1}{2}$
further solving we have $4\cos {60^0}\sin {30^0} = 1$ (by the canceling equal terms using the division)
Hence we get $4\cos {60^0}\sin {30^0} = 1$ which is the required answer.

Note:
In total there are six trigonometric values are sine, cos, tan, sec, cosec, cot while all the values have been relation like $\dfrac{{\sin }}{{\cos }} = \tan $and $\tan = \dfrac{1}{{\cot }}$
And also, we know that the relation of the sine, cosine, and tangent is $\dfrac{{\sin \theta }}{{\cos \theta }} = \tan \theta $ and hence we get the tangent table as

Angle in degrees	\[0^\circ \]	\[30^\circ \]	\[45^\circ \]	\[60^\circ \]	\[90^\circ \]
$\tan $	\[0\]	\[\dfrac{1}{{\sqrt 3 }}\]	\[1\]	\[\sqrt 3 \]	undefined